IMES DISCUSSION PAPER SERIES

INSTITUTE FOR MONETARY AND ECONOMIC STUDIES

BANK OF JAPAN

2-1-1 NIHONBASHI-HONGOKUCHO

CHUO-KU, TOKYO 103-8660

JAPAN

You can download this and other papers at the IMES Web site:

http://www.imes.boj.or.jp

Do not reprint or reproduce without permission.

Comparative Analysis of Zero Coupon Yield Curve Estimation Methods Using JGB Price Data

Kentaro Kikuchi and Kohei Shintani

Discussion Paper No. 2012-E-4

NOTE: IMES Discussion Paper Series is circulated in

order to stimulate discussion and comments. Views

expressed in Discussion Paper Series are those of

authors and do not necessarily reflect those of

the Bank of Japan or the Institute for Monetary

and Economic Studies.

IMES Discussion Paper Series 2012-E-4 April 2012

Comparative Analysis of Zero Coupon Yield Curve Estimation Methods Using JGB Price Data Kentaro Kikuchi* and Kohei Shintani**

Abstract This paper conducts a comparative analysis of the diverse methods for

estimating the Japanese government bond (JGB) zero coupon yield curve (hereafter, zero curve) according to the criteria that estimation methods should meet.

Previous studies propose many methods for estimating the zero curve from the market prices of coupon-bearing bonds. In estimating the JGB zero curve, however, an undesirable method may fail to accurately grasp the features of the zero curve. In order to select an appropriate estimation method for the JGB, we set the following criteria for the zero curve: (1) estimates should not fall below zero, (2) estimates should not take abnormal values, (3) estimates should have a good fit to market prices, and (4) the zero curve should have little unevenness. The method which meets these criteria enables us to estimate the zero curve with a good fit to the JGB market prices and a proper interpolation to grasp the features of the zero curve.

Based on our analysis, we conclude that the method proposed in Steeley [1991] is the best in light of the criteria for the JGB price data. In fact, the zero curve based on this method can fully capture the characteristics of the JGB zero curve in a prolonged period of accommodative monetary policy.

Keywords: coupon-bearing government bond; zero coupon yield; piecewise polynomial function

JEL classification: C13, C14, G12

* Deputy Director and Economist, Institute for Monetary and Economic Studies, Bank of Japan (E-mail: kentarou.kikuchi@boj.or.jp) ** Economist, Institute for Monetary and Economic Studies, Bank of Japan (E-mail: kouhei.shintani@boj.or.jp)

The authors would like to thank Yukio Muromachi (Tokyo Metropolitan University), the participants in the JAFEE 35th Conference, the participants in the Tokyo Metropolitan University Finance Seminar, and staff of the Bank of Japan for their useful comments. We would also like to thank Nikkei Media Marketing, Inc. for granting consent for the publication of zero curve data estimated using JGB price data sourced from that company’s NEEDS service as a supplement to this paper. All the remaining errors are our own. Views expressed in this paper are those of the authors and do not necessarily reflect the official views of the Bank of Japan. The Bank of Japan and the authors do not take any responsibility for any actions by users of the supplemental data.

Table of Contents

1. Introduction ............................................................................................................................ 1 2. Zero Curve Estimation Methods ............................................................................................ 4

A. Basic Items Concerning Interest Rates .................................................................................... 4 B. Undesirable Zero Curves ......................................................................................................... 6 C. Definitions of Notations .......................................................................................................... 8 D. Representative Estimation Methods Proposed in Previous Studies ....................................... 10 E. Degree of Freedom and Locality of the Estimation Methods ................................................ 22

3. Estimation Method Selection Criteria and Comparison Results .......................................... 27 A. Characteristics of JGB Interest Rate Term Structure ............................................................. 27 B. Estimation Method Selection Criteria ................................................................................... 30 C. Comparative Analysis of Estimation Methods ...................................................................... 31

4. Characteristics of the Steeley [1991] Method Modeling Discount Rates ............................ 39 A. Comparison of the Steeley Model with the NS Model .......................................................... 39 B. Comparison of the Steeley Model with Smoothing Spline Methods that Can Estimate

Smooth Instantaneous Forward Rates ................................................................................... 43 5. Conclusion ........................................................................................................................... 45 Appendix 1. Market Conventions for Calculation of JGB Theoretical Prices ...................... 47

A. Definition of Terms ............................................................................................................... 47 B. Market Conventions for Calculating the Number of Days until Cash Flow is Paid .............. 48 C. Response to Changes in Legal and Market Systems ............................................................. 49 D. Initial Interest Payment and Accrued Interest Calculation Methods ..................................... 51

Appendix 2. Estimation Algorithms for the Steeley [1991] Method ..................................... 54 References .................................................................................................................. 57

1

1. Introduction

The zero coupon yield (hereafter, zero yield) is defined as the yield to maturity of

discount bonds. This is used for calculating the present value of future cash flow at any

point of time. The zero coupon yield curve (hereafter, zero curve) which connects zero

yields with different maturities enables us to calculate the present value of any financial

product. Furthermore, using this, we can conduct a comparative analysis between interest

rates with different maturities. Therefore, the zero curve is essential for researchers,

analysts, policymakers and other users.

If bonds with all maturities are traded, we can calculate the zero curve from their market

prices. However, this is usually not the case in the actual market. Therefore, we have to

estimate zero yields for maturities of bonds traded in the market and interpolate them to

obtain zero yields for maturities of bonds which are not traded.1 If discount bonds are

traded, then zero yields with their maturities can be directly derived from their market

prices, and thus the remaining problem is how zero yields for maturities of non-traded

bonds are interpolated. However, in the Japanese Government Bond (JGB) market, there

are few discount bonds with remaining maturities of a year or more. Thus, it is difficult to

derive the JGB zero curve from the market prices of discount bonds,2 and the zero curve

must be estimated from the market prices of coupon-bearing government bonds, which

have a greater variety of maturities.

Diverse methods for estimating the zero curve from the market prices of coupon-bearing

1 In Japan, discount bonds presently issued by the government have a maturity of one year or less. However, the Act on Book-Entry Transfer of Company Bonds, Shares, etc. permits separation of the principal and interest portions of all fixed-interest government bonds (excluding index-linked government bonds and government bonds for individuals) issued after January 27, 2003 (makes these bonds “strippable”), making it possible to trade discount government bonds with a maturity of one year or more on the secondary market. However, we have to note that the liquidity of that market is poor compared with the fixed-interest government bond market. 2 JGB interest rate data can be obtained from information vendors and the Ministry of Finance homepage. However, as far as the authors know, the methodology for the zero curve estimates provided by the information vendors have not been disclosed. Moreover, the interest rates released by the Ministry of Finance are the yields to maturity of fixed-interest government bonds on a semiannual compounded basis, not the zero yield. In addition to these sources, some researchers release data used for their research on their homepages. For example, Johns Hopkins University Professor Jonathan Wright releases zero coupon curve estimates for each country including Japan on a monthly basis on his homepage. He notes that these estimates are based on the Svensson [1995] method handled in this paper, but the data sources are not identified.

2

government bonds have been proposed in previous studies. Representative methods include

(1) the piecewise polynomial method (McCulloch [1971, 1975], Steeley [1991], etc.) which

models the discount function with piecewise polynomials, (2) the non-parametric method

(Tanggaard [1997], etc.) which does not assume any specific structure for the discount

function, (3) the polynomial method (Schaefer [1981], etc.) which models the discount

function with polynomials, and (4) the parsimonious function method (Nelson and Siegel

[1987], Svensson [1995], etc.) which assumes specific functional forms for the zero yield or

instantaneous forward rate. In utilizing the zero curve for an analysis on interest rates, the

zero curve is estimated by one of these four estimation methods. BIS [2005] summarizes

the methods used by the central banks in estimating their government bond zero curves, and

each central bank uses one of the above methods.

When we estimate the JGB zero curve, selecting a method without careful consideration

might result in the estimation of a curve that does not grasp the characteristics of the JGB

yield curve. Moreover, research and analysis using such a zero curve could lead to wrong

conclusions. In order to avoid such problems and select an estimation method that can

accurately grasp the characteristics of the JGB yield curve, this paper compares several

estimation methods proposed in previous studies.

Prior papers that compare multiple zero curve estimation methods include Ioannides

[2003] for U.K. government bonds and Kalev [2004] for Australian government bonds.

However, good estimation methods in previous studies may not necessarily be good for the

JGB market since developments in government bond market prices and market practices

differ from country to country. Previous studies and surveys on the estimation of the JGB

zero curve include Komine et al. [1989], Oda [1997], Inui and Muromachi [2000], and

Kawasaki and Ando [2005]. However, only Komine et al. [1989] compare multiple

estimation methods. Since that paper uses JGB price data from the second half of the 1980s

to compare five estimation methods, we have to note that there is a great difference

between the JGB market environments in the 1980s and since the late 1990s.3 Accordingly,

3 In addition to differences in the interest rate term structure itself including the level and the curve shape, as a market practice, there was the so-called benchmark issue in the 1980s. There was intensive trading of the benchmark issue, while there were few trades of other issues. The concentration of trading on the benchmark issue declined from the mid-1990s, and the benchmark issue disappeared in the late-1990s.

3

using JGB price data from 1999 to 2010, we compare the representative estimation methods

proposed in previous studies and select an estimation method that can grasp the

characteristics of the JGB yield curve.

For JGB yield curves since 1999, yield curves under the zero interest rate policy and the

quantitative easing policy are distinctive. As a feature of yield curves during these periods,

we can point out that the yield curve has a flat shape near zero at the short-term maturities.

Some estimation methods cannot grasp this kind of curve shape, and sometimes estimate

zero yields below zero. Thus, it is necessary to compare the estimation methods for the JGB

zero curve. In this paper, we set the criteria to make an appropriate selection of the

estimation method. We then select the optimal estimation method based on these criteria.

This approach has not been tried in previous studies. Specifically, we first reject

inappropriate methods based on the following criteria: (1) zero yield estimates should not

fall below zero and (2) zero yield estimates should not take abnormal values. Next, from

among the remaining estimation methods, the most desirable method is chosen based on the

criteria of (3) good fit to JGB market prices and (4) little unevenness in the zero curve. As a

result of this selection process, the estimation method in Steeley [1991] is selected.

The remainder of this paper is organized as follows. Section 2 explains the zero curve

estimation methods. Section 3 presents the selection criteria for the zero curve estimation

methods and uses these criteria to compare several representative estimation methods

proposed in the prior studies. Section 4 clarifies the characteristics of the method in Steeley

[1991] when applied to the JGB market through comparisons with other estimation

methods. Section 5 presents our conclusions.

As reference materials so that readers can reproduce the estimation methods, Appendix 1

summarizes the JGB market conventions required to calculate theoretical JGB prices such

as the definition of the timing of when cash flows are paid, the method of calculating the

number of days until when cash flow is paid, and the method of calculating accrued interest.

Appendix 2 explains the details of the estimation algorithm in Steeley [1991]. As a

supplement, we also attach the daily JGB zero curve data from January 1999 through

December 2011 estimated using the method in Steeley [1991].4 4 The data can be obtained from http://www.imes.boj.or.jp/research/papers/english/12-E-04.txt.

4

2. Zero Curve Estimation Methods

In this section we first define the zero yield and the zero curve, and summarize other

basic items regarding interest rates required in this paper. Next, as a premise for considering

the suitability of the estimation methods, we summarize the characteristics of undesirable

zero curves. We then explain representative methods for the zero curve estimation and

examine the characteristics of each method based on the two concepts of “degree of

freedom” and “locality” from the perspectives of not estimating an undesirable zero curve

and estimating a zero curve that accurately expresses market prices.

A. Basic Items Concerning Interest Rates

We define the present time as t and the present value of bonds called discount bonds that

will certainly pay cash flow 1 at the future time T as ( , )Z t T . The zero yield ( , )y t T from

t to T is defined as the yield to maturity of the discount bond.

)).,(log(1),( TtZtT

Tty�

�� (1)

This paper estimates zero yields for all maturities since we anticipate the potential use of

zero yields for comparative analysis between yields with different maturities, as well as the

use of the zero yield of a specific maturity. In other words, we estimate the curve

connecting the zero yields with different maturities, which is called the zero curve.

Specifically, we describe the zero curve at time t as a function of the remaining maturity x,

),,( xtty � (2)

and estimate the curve ),( xtty � for x at time t. As explained in Section 2D below, in

previous studies, the discount rate ),( xttZ � is modeled through functional form, etc.

Hereafter ),( xttZ � is sometimes referred to as the discount function, as a function of x.

With the zero curve, it becomes possible to calculate the instantaneous spot interest rate,

that is, the instantaneous interest rate at time t. The instantaneous spot interest rate ( )r t at

5

time t is defined as Equation (3).

).,(lim)(0

xttytrx

���

(3)

Furthermore, with the zero curve it also becomes possible to calculate the implied

forward rate. The implied forward rate is the interest rate over a period from a future point

in time, defined at the present time. Specifically, the implied forward rate from time S to

time T (S < T) at t is defined so that the value derived when the cash flow 1 at T is

discounted through S by the implied forward rate and then discounted by the zero yield

from t to S is equal to the present value when the cash flow 1 at T is discounted by the zero

yield from t to T. Therefore, the implied forward rate ),,( TStf from S to T at t is defined

by Equation (4).

)).)(,(exp()))(,,(exp(),( tSStySTTStfTtZ ����� (4)

From Equation (1) and Equation (4), the implied forward rate ),,( TStf is calculated as

Equation (5) using the discount bond price at t.

.),(),(log1),,( ���

�

��

��StZTtZ

STTStf (5)

Additionally, the instantaneous forward rate at t for S as seen from t, ),( Stf is defined as

shown in Equation (6).

)).,(log(),(),(log1lim),,(lim),( StZ

SStZTtZ

STTStfStf

STST ��

�����

�

��

�����

(6)

From Equation (6) and 1),( �SSZ , the following equation expresses the relation between

the discount function ),( xttZ � and the instantaneous forward rate.

.),(exp),(0

���

� ����

xdssttfxttZ (7)

Furthermore, from Equation (1) and Equation (7), the relation between the zero yield and

the instantaneous forward rate is as follows.

6

.),(1),(0 ���x

dssttfx

xtty (8)

B. Undesirable Zero Curves

Estimation methods that can fully grasp the market prices of the bonds are preferable for

estimating zero curves. If a zero curve is derived with an estimation method that does not

adequately fit the market prices, there are concerns that the zero curve might not fully

reflect the information contained in the prices. Using such zero curves in interest rate

analyses may lead to erroneous conclusions. On the other hand, methods selected based

solely on the good fit to the market prices could result in an undesirable zero curve with

improper interpolations. We now show the types of undesirable zero curves that may be

estimated.

Figure 1: Zero curve that does not meet the zero interest rate constraint (conceptual diagram)

Remainingmaturity

Zero

(1) Violation of the Zero Interest Rate Constraint

Because the estimated zero curve is the nominal interest rate, zero curves falling below

zero for some maturities like the curve in Figure 1 are considered to be undesirable.

7

(2) Excessive Unevenness in the Zero Curve

Even for estimation methods with zero curves that do not go below zero, the zero curve

may have an excessively uneven shape due to the characteristics of the estimation method.

Figure 2: Unevenness of the zero curve (conceptual diagram)

10 year

A B

9 year 11 year RemainingMaturity

Notes Zero yield calculated from discount bond market prices Zero yield estimate calculated using estimation method A Zero yield estimate calculated using estimation method B

Figure 2 is a conceptual diagram which shows the differences in the unevenness of the

zero curves estimated using two estimation methods. In Figure 2, we assume that discount

bonds with some maturities are traded on the market and zero curves are estimated based

on those discount bond prices. Figure 2 shows that the two curves based on estimation

method A and B both fit the market prices very well. However, method B has far greater

unevenness. It is likely that curve B is unreasonable from the principle of the zero curve

characteristic of little variation in zero yields with proximate maturities. The excessive

unevenness of this type of zero curve, rather than reflecting the information contained in

bond prices, may result from the peculiar characteristics of the estimation method. Thus

conducting analyses using a zero curve with very great unevenness like curve B could lead

to erroneous results. For that reason, estimation methods with great unevenness in the zero

curve like method B in Figure 2 cannot be considered desirable.

8

(3) Abnormal Values

Some methods estimate zero yields with excessively high or excessively low values that

are abnormal. Either those kinds of the estimation methods have a poor fit with bond

market prices due to their weak expressive power, or they have an over-fit with the market

prices.

Figure 3: Zero curve with abnormal values (conceptual diagram)

Remainingmaturity

Figure 3 is a conceptual diagram presenting a zero curve with abnormal values. The

curve in this figure overestimates the interest rate for some short-term maturities. This type

of problem results from the characteristics of the estimation method and does not reflect the

information contained in the bond prices.

C. Definitions of Notations

Before explaining the contents of previous studies in Section 2D, we prepare the

notations used in this paper.

For the sake of simplicity hereafter, the date when the zero curve is estimated is set at

0�t and each point in time is expressed as the number of years from 0�t . The discount

function ),0( xZ , the zero curve ),0( xy and the instantaneous forward rate ),0( xf are

abbreviated as )(xZ , )(xy , and )(xf .

9

Next, we define the notations concerning fixed-coupon bearing bonds traded on the

market. All bonds issued on or prior to the estimation date ( 0�t ) are expressed as

}),,1{( namenii �� with the notional amount of bond i expressed as iN and the coupon

rate (annual rate) as ic . All the times when bond i generates cash flow are expressed in

terms of the number of years from 0�t , as },,{ 1i

nii

icf

TT ��T . Here icfn represents the

number of times that bond i's cash flows are paid after 0�t . We assume that if lk � , then i

li

k TT � . Additionally, T is defined as the union of the iT of all the bonds issued at or

before 0�t as follows.

.min},;{min},,,{:1

},,1{11

1},,1{1

1

ik

nknij

ik

ik

nknijn

n

i

i TTTTTTTTicf

nameicf

namecf

name

���

�

����

�������

�� TT

We also define the notations regarding bonds traded on the market at the present time

( 0t � ). First, the bonds traded on the market at the present time are expressed as

},,{ 1 Invv ��I . Then the present market price of each bond is expressed as T),,( 1 Invv PP ��P where the price is the bare value, and the accrued interest at execution

of transactions is denoted by T),,( 1 Invv AA ��A . 5 The superscript T indicates a

transposition of a vector or a matrix (here and hereafter, except as otherwise noted). For the

details of the accrued interest calculation method, see Appendix 1D(2).

Finally, we prepare the notations used to express the theoretical price of bond iv on the

date which the zero curve is estimated. First we define the vector ivc concerning the cash

flow of bond iv as follows;6

5 We note that P and A are dependent on the present time. Here, for simplicity, the present time 0�t is omitted. The model parameter α introduced hereafter is also a variable dependent on the present time. 6 The cash flow from fixed-coupon bonds issued since March 2001 are handled in this way, while a slightly different form is used for fixed-coupon bonds issued before March 2001. See Appendix 1D(1) for the details.

10

� �

,

otherwise0

if2

,if2

),,(

,),,(,),,,(,),,,(T

1

���

�

���

�

�

��

��

�

�

iiv

cfni

ii

iiv

cfni

ii

ii

cf

iiiiiii

vj

vvv

vj

vj

vv

jvv

nvv

jvvvvv

TTNNc

TTTNc

TNcg

TNcgTNcgTNcg

T

c ��

where ivc is a vector of 1�cfn . If the jth element of ivc is expressed as ivjc , the

theoretical price ivQ of bond iv on the estimation date is as shown in Equation (9).

.)(1

i

cf

ii vn

jj

vj

v ATZcQ ����

(9)

The next subsection 2D presents an outline of the representative previous studies on zero

curve estimation. Previous studies all directly or indirectly model the discount function

)(xZ as a function of parameter α . To emphasize this point, we sometimes express the

discount function )(xZ as );( αxZ . Since the discount function depends on parameter α ,

the theoretical price of each bond also depends on parameter α through Equation (9).

Thus, we express the theoretical price of bond iv as )(αivQ and write the theoretical

prices of all bonds traded at 0�t as the vector form T))(,),(() 1 ααQ(α Invv QQ �� .

D. Representative Estimation Methods Proposed in Previous Studies

In estimating the zero curve from the market prices of bonds, how we model the discount

function ( )Z x is important. All previous studies which model the discount function ( )Z x

adopt one of the following four methods: (1) the piecewise polynomial method, (2) the

non-parametric method, (3) the polynomial function method, and (4) the method which

assumes a specific functional form for the discount function. In this subsection, we present

an outline of all the estimation methods which we deal with later in Section 3C.

In estimating the zero curve, in addition to modeling the discount function ( )Z x , it is

11

also necessary to set the objective function in order to determine the parameters. In some

previous studies, the objective function taking the weighted residual sum of squares is used,

instead of the simple residual sum of squares of errors between the market prices and the

theoretical prices based on the model.

For example, according to BIS [2005], the Bank of Canada and the Bank of Spain adopt

the objective function taking the residual sum of squares weighted by the inverse of the

bond durations. That objective function places an emphasis on the fit to the prices of

short-term bonds over long-term bonds. However, since we use fixed-coupon JGBs (2 year,

5 year, 10 year, 20 year, and 30 year) for the zero curve estimation, there are more

short-term bonds than long-term bonds. Therefore, using the residual sum of squares

weighted by the squares of the inverse of the durations as the objective function might

result in the under-fitting to the market prices of long-term JGBs. Accordingly, we adopt

the objective function with the simple residual sum of squares without taking any weights,

as in McCulloch [1975], Steeley [1991] and many other previous studies. As Inui and

Muromachi [2000] point out, adopting the simple residual sum of squares as the objective

function may lead to heterogeneity of variance of residuals. Taking the weighted residual

sum of squares of errors as the objective function is a solution to this problem. However,

depending on the setting of weights, heterogeneity of variance of residuals does not always

disappear; moreover, the estimation may result in a poor fit to a bond price. Considering

these points, in this paper, we assume the simple residual sum of squares as the objective

function.

Furthermore, some of the previous studies adopt a function that adds a penalty term

concerning the curvature of the instantaneous forward rate term structure to the residual

sum of squares as the objective function, in order to estimate a zero curve with a smooth

instantaneous forward rate term structure (Fisher, Nychka and Zervos [1995], Waggoner

[1997], Jarrow, Ruppert and Yu [2004], etc.). This is called the smoothing spline method.

While the zero curve estimated using such an objective function has a smoothed forward

rate term structure, smoothing results in less fit to market prices. Additionally, in the

smoothing method, there remains arbitrariness in selecting criteria determining the level of

smoothness. Therefore, we exclude the smoothing spline method from the several

12

estimation methods analyzed in Section 3. We select the models which estimate

comparatively smooth zero curves without smoothing.

Considering the above, in this section, we assume the objective function is the simple

residual sum of squares.

(1) The Piecewise Polynomial Method7

We here introduce previous studies which model the discount function by using the

piecewise polynomial. As means of modeling the discount function, most of all previous

studies directly model the discount function or indirectly model the discount function

through modeling the instantaneous forward rate.

First, we define the piecewise polynomial function. For this purpose, we have to set the

sequence of points known as knot points. The knot points are the following sequence,

nnmm uuuu ���� �� 11 � ,

where m and n are integers. When the knot points are given, for integer j, the piecewise

polynomial function of degree l , ( , )B j x is continuous with respect to the real number x,

and is the polynomial function on )1(],[ 1 ���� nhmuu hh , ],( mu�� , and ),[ �nu .

Methods Directly Modeling the Discount Function

(i) The McCulloch [1975] Method

McCulloch [1975] models the discount function ( )Z x as a linear combination of

piecewise polynomial functions. First, in this method, the knot points are set as

knotnuuuuu ������ � �21010 . Then McCulloch [1975] defines the piecewise

polynomial ( , ) ( 0, , )knotB k x k n� , )knot, as the third-degree piecewise polynomial shown

below in Equation (10).

7 Many of previous studies model the discount function using piecewise polynomials known as spline

functions. Hence, the piecewise polynomial method is also called the spline function method. The spline function of degree l is the piecewise polynomial whose derivatives from the first order to the l-1th order are all continuous.

13

.),(,For

,),26

2)((

,,

)(6)(

2)(

2))((

6)(

,,)(6

)(,,0

),(

,For

1111

11

1

21

32

12

1

11

31

1

xxkBnk

xuuxuuuuu

uxu

uuuxux

uxuuuu

uxuuu

uxux

xkB

nk

knot

kkkkk

kk

kk

kk

kk

kkkkk

kkkk

k

k

knot

��

�����

�

�����

�

�

��

���

�

��

��

��

�

���

�

���

��

�

�

����

��

�

��

��

��

�

�

(10)

McCulloch [1975] expresses the discount function ( )Z x as a linear combination of

piecewise polynomials defined in Equation (10) as follows,

.),(1)(0

��

��knotn

kkxkBxZ � (11)

From the characteristics of the discount function, (0) 1Z � must hold true. Modeling the

discount function as in Equation (11) is consistent with 1)0( �Z because

),,0(0)0,( knotnkkB ��� from Equation (10).

When we substitute Equation (11) into Equation (9), the theoretical price ivQ for bond

iv is expressed as a function of parameter T10 ),,,(

knotn��� ��α as follows,

,)(),()( T

10 11Bαcα i

cf

iknot cf

i

cf

ii vn

j

vj

n

kk

n

jj

vj

n

j

vj

v cTkBccQ ����

�

���

��

�

���

��

�

�� �� ��

�� ��

� (12)

where B is the )1( �� knotcf nn matrix with ),( jTkB as the ),( kj th element. ivc is the

1�cfn vector with ivjc as the j th element.

As mentioned at the beginning of this section, the zero curve estimation in this paper

uses the residual sum of squares of errors between bond market prices and theoretical prices

as the objective function. In this case the estimated value α̂ of parameter α is obtained

as the solution to the following optimization problem.

14

� � � �

.))(,,)((:)(~

,),,(:~

,)(~~)(~~minargˆ

T

11

T

11

T

11

11

��

��

��

��

���

���

���

!" ���

cfnn

cf

cfnn

cf

n

j

vj

vn

j

vj

v

n

j

vj

vn

j

vj

v

II

II

cQcQ

cPcP

αααQ

P

αQPαQPαα

�

� (13)

Since )(~ αQ is a linear function of α , Equation (13) can be regarded as a least squares

optimization problem. Thus, the optimal solution α̂ for parameter α is obtained as

Equation (14),

,~)())((ˆ T1T PBcBcBcα �� (14)

where c is set with T),,( 1 Invv ccc �� and it is the cfI nn � matrix. 1�X is the inverse

matrix of the square matrix X .

Hereafter, this method is sometimes referred to as the “McCulloch [1975] Method

Modeling Discount Rates.”

(ii) The Steeley [1991] Method

Steeley [1991], like McCulloch [1975], represents the discount function ( )Z x as a

linear combination of piecewise polynomial functions. However, there is a difference in

that Steeley [1991] models the discount function ( )Z x as shown in Equation (15).

.),()(1

3�

�

��

�knotn

kkxkBxZ � (15)

Steeley [1991] also proposes a different functional form than McCulloch [1975] for the

piecewise polynomial function ( , )B k x . Steeley [1991] sets the knot points as

3213 ���� �����knotknotknotknot nnnn uuuuu � . Then ( , )B k x is recursively defined as shown in

Equation (16).

15

.1for),(),1(),(),(

,otherwise0

,1:),(),(

11

11

11

��

���

��

��

��� ��

��

���

���

�

�

DxkBuu

uxxkBuu

xuxkBxkB

uxuxkBxkB

DkkD

kD

kkD

kDD

kk

(16)

The function in Equation (16) is called a B-spline function. Steeley [1991] proposes a

model based on a piecewise cubic polynomial with 4D � in Equation (16). Here, we note

that the following equation holds true from Equation (15) and (0) 1Z � .

.1)0,(1

3��

�

��

knotn

kkkB � (17)

When we substitute Equation (15) into Equation (9), the theoretical price ivQ for bond

iv is represented as follows,

,)(

),(),()(

T

1

3 11

1

3

Bαc

α

i

knot cf

i

cf knotii

v

n

kk

n

jj

vj

n

j

n

kkj

vj

v TkBcTkBcQ

�

���

�

���

��

�

�� � �� �

�

�� ��

�

��

�� (18)

where B is the )3( �� knotcf nn matrix with ),( jTkB as the ),( kj th element.

From Equation (17) and Equation (18), parameter α is estimated as the solution to the

following constrained least squares optimization problem.

� � � �# $

.1)0,(..

,)()(minargˆ

1

3

T

�

���

��

��

knotn

kkkBts �

αQPαQPαα

(19)

Solving this, the optimal solution α̂ is as shown in Equation (20),

,))(())((

)())((1)())((ˆ 01T

01T

0

T1T0T1T BBcBc

BBcBcBPBcBcBcBPBcBcBcα �

�

�� �

�� T

T (20)

where T0 ))0,1(,),0,3(( ��� knotnBB �B .

16

Hereafter, this method is sometimes referred to as the “Steeley [1991] Method Modeling

Discount Rates.” The details of the estimation algorithm for this method are presented in

Appendix 2.

Methods Modeling Instantaneous Forward Rates

(iii) The Fisher, Nychka and Zervos [1995] Method

Fisher, Nychka and Zervos [1995] represent the zero curve estimation methods using

piecewise polynomial functions in the general form to model the discount function )(xZ ,

the zero yield )(xy and the instantaneous forward rate )(xf , respectively. In this paper,

among these we explain the case when )(xf is modeled directly.8

Fisher, Nychka and Zervos [1995] model the instantaneous forward rate )(xf as a

linear combination of piecewise polynomial functions as follows,

,),()( ��

�n

mkkxkBxf � (21)

where ),( xkB is a piecewise polynomial.

Using Equation (7) and Equation (21), the discount function can be expressed as follows.

.),(:),(

,),(exp)),((exp

),(exp)(exp)(

0

0

00

��

�

�

���

�

�����

�

�

���

���

�

����

��

���

��

�

x

k

n

mkk

n

mk

x

x n

mkk

x

dsskBxkB

xkBdsskB

dsskBdssfxZ

��

�

(22)

When we substitute Equation (22) into Equation (9), the theoretical price ivQ can be

8 The primary objective in Fisher, Nychka and Zervos [1995] is to estimate a smooth instantaneous forward rate term structure, and their estimate is conducted with smoothing. In our paper, however, as stated at the beginning of this section, the goal is to estimate a zero curve that simultaneously achieves a good fit with market prices and an appropriate interpolation. Therefore, our paper does not place emphasis on making the instantaneous forward rate term structure smooth. In this respect, the Fisher, Nychka and Zervos [1995] model explained here has the different objective function from their research. We advance our discussion using the simple sum of the squares of the residuals of the theoretical prices and the market prices as the objective function.

17

written as follows,

,)),(exp(,),),(exp(:)exp(

),exp()(),(exp)(

T

1

T

1

���

�

�����

�����

�

���

��

� �

��

� �

n

mkkn

n

mkk

vn

jk

n

mkj

vj

v

cf

i

cf

ii

TkBTkB

TkBcQ

��

�

�αB

αBcα

(23)

where ),(, jkj TkB�B .

Because )(αivQ is a nonlinear function of α , it is necessary to solve a nonlinear

optimization problem in order to find the optimal solution for α . Accordingly, Fisher,

Nychka and Zervos [1995] simplify the optimization problem by conducting a first order

Taylor approximation around an arbitrary point 0αα � in Equation (23). Specifically,

)(αivQ is approximated as follows,9

# $ ),())(exp(*)()(

),()exp()()(

)),()exp()(exp()()(

0T0T0

0T0

00T

0

0

αα1αBBcα

αααBα

cα

αααBα

αBcα

αα

αα

����

����

��

����

��%

�

�

ii

ii

ii

vv

vv

vv

Q

Q

Q

(24)

where 1 is the ( 1)knotn � dimensional vector of T(1,...,1)�1 . The asterisk * in

Equation (24) represents the element-by-element product of vectors or matrices. With the

following Equation (25),

� �# $

,)()()(

,)exp(*)()(0000

T0T0

ααXααY

1αBBcαXiiii

ii

vvvv

vv

QP ���

��� (25)

the optimal solution 0ˆ ( )α α based on an approximation of Equation (23) around an

arbitrary point 0�α α is the solution of the following optimization problem.

9 There are errors in the formula equivalent to Equation (24) in the original paper.

18

� � � �

.))(),...,((:)(

,))(),...,((:)(

,)()()()(minarg)(ˆ

T000

T000

00T000

1

1

αYαYαY

αXαXαX

ααXαYααXαYααα

I

I

n

n

vv

vv

�

�

���

!" ���

(26)

There is no need to consider the constraint condition (0) 1Z � on the discount function for

the optimization problem in Equation (26) because the following equation holds from

Equation (22).

.10exp)0,(exp)0( ���

�

�����

�

�

��� ��

��

n

mkk

n

mkkkBZ ��

Equation (26) can be solved as an unconstrained least squares problem as follows.

� � ).()()()()(ˆ 0T010T00 αYαXαXαXαα�

� (27)

However, this solution depends on the point where the Taylor approximation is conducted, 0αα � . Thus, Fisher, Nychka and Zervos [1995] conduct a Taylor approximation of

)(αivQ around the point 10)(ˆ ααα & like that in Equation (24), and calculate the optimal

solution 21)(ˆ ααα & like in Equation (27). They also repeat the same operation for 2α

and propose the convergence point of the optimal solution )(ˆ iαα as the optimal solution

of the parameter.

While this is the estimation method proposed in Fisher, Nychka and Zervos [1995], in

implementing the estimation under this method, the piecewise polynomial function ),( xkB

in Equation (21) must be determined. In the choice of estimation methods in Section 3C

below, the methods considered for selection include (1) the method using the piecewise

quadratic polynomial10 proposed in McCulloch [1971] as ),( xkB (hereafter, sometimes

referred to as the “McCulloch [1971] Method Modeling Instantaneous Forward Rates”) and

(2) the method using the B-spline function of degree two in Steeley [1991] as ),( xkB

(when D=3 in Equation (16)) (hereafter, sometimes referred to as the “Steeley [1991]

Method Modeling Instantaneous Forward Rates”). Like the cases when the discount

10 This is the differential function with respect to x of Equation (10) (see McCulloch [1971]).

19

function is directly modeled in D(1)(i) and (ii) of this section, the discount functions for

both (1) and (2) are cubic functions.

(2) The Non-parametric Method

(i) The Tanggaard [1997] Method

Unlike the above approaches, Tanggaard [1997] does not use a piecewise polynomial

function to represent the discount function, but rather deals with the discount function of

each term as a parameter.11 In other words, Tanggaard [1997] represents the theoretical

price )(αivQ as follows.

T

1( ) , ( ).

cf

i i i

nv v v

j j j jj

Q c Z T� ��

� � �� c α (28)

Because the theoretical price )(αivQ is a linear function of α , the optimal solution α̂

can be obtained by solving Equation (29).

# $.))(())((min T αQPαQPα

�� (29)

Solving this, the optimal solution α̂ is as shown in Equation (30),

,)(ˆ T1T Pcccα �� (30)

where, T),,( 1 Invv ccc �� .

(3) The Polynomial Method

(i) The Schaefer [1981] Method

Schaefer [1981] represents the discount function ( )Z x using a linear combination of

polynomials known as Bernstein polynomials. The Bernstein polynomial ),( xkBD of

degree D is the polynomial defined as follows. 11 Aside from Tanggaard [1997] introduced here, other non-parametric methods include Carleton and Cooper [1978] and Houglet [1980].

20

.1),0(

,!)!(

)!(:

,0,)1(),(0

1

�

���

����

�

� �

����

�

�

� ���

��

�

��

xBjjkD

kDj

kD

kjk

xj

kDxkB

D

jkkD

j

jD

(31)

Schaefer [1981] represents the discount function as a linear combination of the Bernstein

polynomials as follows.

.),()(0

��

�D

kkD xkBxZ �

Since 1)0( �Z and ,0for0)0,( �� kkBD we obtain 0 1� � from the above equation.

Therefore, )(xZ is represented in the following form.

.),(1),()(10

����

���D

kkD

D

kkD xkBxkBxZ �� (32)

When we substitute Equation (32) into Equation (9), the theoretical price ivQ of bond

iv is expressed as a function of the parameter 1( , , )TD� ��α , )TD, as follows,

,)(),()( T

11 11Bαcα i

cf

i

cf

i

cf

ii vn

j

vj

D

kk

n

jjD

vj

n

j

vj

v cTkBccQ ����

�

���

��

�

���

��

�

�� �� ��

�� ��

� (33)

where B is the cfn D� matrix with ),( jD TkB as the ),( kj th element.

From Equation (33), the optimal parameter α̂ can be obtained by solving the following

least squares problem.

� � � �# $

).)(,...,)((:)(~

,),...,(:~

,)(~~)(~~minargˆ

11

T

11

T

11

11

��

��

��

��

���

���

���

cfnn

cf

cfnn

cf

n

j

vj

vn

j

vj

v

n

j

vj

vn

j

vj

v

II

II

cQcQ

cPcP

αααQ

P

αQPαQPαα

(34)

Thus, the optimal solution α̂ becomes as shown in Equation (35).

21

.~)())((ˆ T1T PBcBcBcα �� (35)

(4) Parsimonious Function Methods

(i) The Nelson and Siegel [1987] Method

Nelson and Siegel [1987] use a parametric function in Equation (36) to model the

instantaneous forward rate )(xf .

.expexp)(33

23

10 ���

�

�����

�

�

����

���

��� xxxxf (36)

From Equation (8) and Equation (36), the zero curve )(xy is calculated as follows,

.exp

/)/exp(1

/)/exp(1

)(1)(

33

32

3

310

0

���

�

����

�

���

�����

�

�

� ����

�

����

���� x

xx

xx

dssfx

xyx

(37)

Here, the theoretical price ivQ of bond iv is expressed by Equation (9). Since the

discount function );( αxZ is a nonlinear function of the parameter T3210 ),,,(: �����α

as shown in Equation (37), the optimal solution α̂ is obtained by solving the following

nonlinear optimization problem.

� �' (Tˆ arg min ( ) ( ( )) .� � �α

α P Q α P Q α (38)

(ii) The Svensson [1995] Method

Svensson [1995] adds a new term to the functional form proposed in Nelson and Siegel

[1987] (see Equation (36)) for modeling the instantaneous forward rate to improve the

expressive power of the instantaneous forward rate )(xf . Specifically, )(xf is modeled

using the following functional form.

22

.expexpexp)(55

433

23

10 ���

�

�����

�

�

�����

�

�

����

���

���

��� xxxxxxf (39)

From Equation (39), the zero yield )(xy is calculated as follows.

.exp/

)/exp(1

exp/

)/exp(1/

)/exp(1

)(1)(

55

54

33

32

3

310

0

���

�

����

�

���

���

���

�

����

�

���

�����

�

�

� ����

�

����

����

����

xx

x

xx

xx

x

dssfx

xyx

(40)

From Equation (40), as in the Nelson and Siegel [1987] method, the theoretical price ivQ of bond iv is a nonlinear function of the parameter; therefore, the parameter

estimation is solved using a nonlinear optimization problem.

E. Degree of Freedom and Locality of the Estimation Methods

In subsection B, we presented undesirable zero curves. In this subsection, we introduce

the two concepts, “degree of freedom” and “locality” into the estimation methods to select

methods which estimate desirable zero curves and reject ones which estimate undesirable

zero curves. We then examine each of the zero curve estimation methods based on this

perspective.

We begin with the explanation of the degree of freedom of the estimation method. In

previous studies on the zero curve estimation methods introduced in subsection D, the

discount functions or the instantaneous forward rates are modeled with some functions. In

this paper, we define the degree of freedom as the difference between the number of

parameters of those functions and the number of constrained conditions imposed on those

parameters. For example, the degree of freedom of the McCulloch [1975] Method

Modeling Discount Rates explained in D(1)(i) is equal to the number of knot points + 1,

while the degree of freedom of the Nelson and Siegel [1987] Method in D(4)(i) is equal to

23

4. In general, estimation methods with a low degree of freedom give a poor fit to the market

prices, while methods with a high degree of freedom give a good fit to the prices. However,

as shown in subsection B, estimation methods with an excessively high degree of freedom

may estimate zero curves with excessive unevenness or inappropriate interpolations. In

order to avoid such an undesirable zero curve, it is necessary to use an estimation method

without an excessively high degree of freedom.

Next, we explain the locality of the estimation method. For the sake of simplicity, we

assume that the discount bonds are traded on the market and the zero curve is estimated

from the discount bond prices. The concept of locality means the extent to which the shape

of the estimated zero curve is changed when a discount bond price changes. In other words,

if a change in the price of a discount bond with a given maturity greatly changes the zero

curve estimates with other maturities, the estimation method is said to have low locality.

Conversely, if the price change of a discount bond does not greatly change the zero curve

estimates for other maturities, the estimation method is said to have high locality. One

advantage of estimation methods with high locality is that even when a bond with a

remaining maturity has an abnormal price, this has almost no effect on the zero curve

estimates for the other maturities. Another advantage of methods with high locality is that

they are likely to estimate complex shaped zero curves with multiple inflection points.

We now formulate the concept of locality described above. The original discount bond

price data are denoted by P , and ivP)�P denotes the price data obtained when only the

price of bond iv with a remaining maturity iiv

cf

vnT changes by ivP) while there are not

any price changes in the other bonds. The zero curve estimated with method X from P

is denoted by );(~ PxyX , and the zero curve estimated from ivP)�P is denoted by

);(~ ivX Pxy )�P . Then, the following index );,( i

ivcf

vnX Tl *) defined in Equation (41) may be

considered as an index to measure the locality for the maturity iiv

cf

vnT of method X ,

,|);(~);(~|

|);(~);(~|:);,(

2

02

�

�

���

���

cfn

iviv

cfn

i

iviv

cfn iiiv

cf

T

T Xv

X

TX

vX

vnX

dxxyPxy

dxxyPxyTl

*

*

)

)*)

PP

PP (41)

where * is a sufficiently small positive real number and ) is a real number. If the value

24

calculated in Equation (41) is small compared with those of the other estimation methods,

at least, it is said that method X has higher locality around the remaining maturity iiv

cf

vnT of

bond iv than the other methods. If the values calculated from Equation (41) for all bonds

(or all maturities) are small compared with those of the other estimation methods, then the

method X has high locality for all maturities.

However, Equation (41) cannot be considered an easy-to-use index because it depends on

the choice of * and ) . In addition, we have to calculate indexes of all bonds to judge the

locality for any maturity. Therefore, as an index to comparatively easily evaluate the

locality of estimation methods, we now consider the ratio of the number of parameters that

contribute to the determination of the discount rate with a remaining maturity to the degree

of freedom. The lower this ratio is, the higher the locality. For example, the ratio for the

Nelson and Siegel [1987] method for any maturity is equal to 1. Next, we calculate the ratio

for the Steeley [1991] Method Modeling Discount Rates in subsection D(1)(ii). If the knot

points are set as )33,,3( ���� llul , the degree of freedom becomes 32. Using this, we

calculate the ratios as follows. The ratio for maturities of less than a year is 125.032/4 � ,

the ratio for maturities of 2-30 years in annual increments is 0625.032/2 � , and the ratio

for all other maturities is 09375.032/3 � . Hence, the Steeley [1991] Method Modeling

Discount Rates has higher locality than the Nelson and Siegel [1987] method for all

maturities.

Next, we see how the locality differs according to the kind of piecewise polynomial

method. The locality of the Steeley [1991] Method Modeling Discount Rates is different

from that of the McCulloch [1975] Method Modeling Discount Rates. Under the

McCulloch [1975] Method Modeling Discount Rates, setting the knot points as

0),30,,0( 1 ��� �ullul � , the degree of freedom is equal to 31. The ratio between the

number of parameters that contribute to the determination of the discount function and the

degree of freedom is 0645.031/2 � for maturities of one year or less, 0968.031/3 � for

maturities of 2 years or less, and 129.031/4 � for maturities of three years or less. In this

way, it rises as the maturities increase. This shows that except for remaining maturities of

one year or less, the Steeley [1991] Method Modeling Discount Rates has lower values than

the McCulloch [1975] Method Modeling Discount Rates. We therefore conclude that the

25

Steeley [1991] Method Modeling Discount Rates has higher locality.12

Figure 4: Characteristics of each method (conceptual diagram)

Parsimo-nious

function

Locality

Degreeof

freedom

High

Poly-

nomial

Piecewise polynomial

Non-parametric

Low

High

Figure 4 arranges the previous studies in subsection D from the viewpoints of the

concepts of degree of freedom and locality. The Tanggaard [1997] method, one of the

non-parametric methods, directly estimates discount rates corresponding to times when all

cash flows of all bonds are paid. Consequently, the degree of freedom of this method is

much higher compared with the others. This method also has extremely high locality

because it models the discount rate without assuming any specific functional form. In

contrast, there are limits to the locality of the polynomial method and the parsimonious

function method because they express the entire term structure of the discount function or

the instantaneous forward rate as a single functional form. As for the degree of freedom, we

can deal with polynomial methods with various degrees of freedom by changing the degree

of the polynomial used in the modeling. This is also true of the degree of freedom of the

parsimonious function method. For the piecewise polynomial method, the degree of

12 Like the McCulloch [1975] method, the McCulloch [1971] method also has lower locality compared with

the Steeley [1991] method.

26

freedom depends on the number of knot points. The degree of freedom of this method rises

as the number of knot points increases; as a result, the ratio of the number of parameters

contributing to the discount rate of a given maturity to the degree of freedom decreases.

This means that the locality of the piecewise method rises as the number of knot points

increases. Thus the locality of the piecewise polynomial method is higher compared with

the localities of the polynomial method and the parsimonious function method.

In the next section we conduct comparative analyses on the eight zero curve estimation

methods introduced in subsection D. Here we briefly explain why we narrowed down the

target to the eight estimation methods from among the many methods discussed in previous

studies. The range of the previous studies on the non-parametric methods, the parsimonious

function methods and the polynomial methods is somewhat narrow. Hence, we have chosen

one or two representative examples of each. On the other hand, various piecewise

polynomial methods have been proposed in previous studies. We include the McCulloch

[1971, 1975] methods and the Steeley [1991] method which have different locality in our

analysis. Since some of the methods with piecewise polynomials model the discount rate as

in subsection D(1)(i) and (ii) and others model the instantaneous forward rate as in

subsection D(1)(iii), we analyze four piecewise polynomial methods based on the

McCulloch [1971, 1975] methods and the Steeley [1991] method. Aside from these

methods, other piecewise polynomial methods include Vasicek and Fong [1982] and

McCulloch and Kochin [2000].13 The localities of these methods are between or below

those of the McCulloch [1971, 1975] methods and the Steeley [1991] method. Accordingly,

we do not deal with any other piecewise polynomial methods in this paper’s analysis aside

from the four methods specified above.

13 Vasicek and Fong [1982] conduct the variable change )exp(1 sx ���� on the discount function )(xZ ,

and then use a piecewise polynomial to model the newly defined function ))(()(~ xZsZ � . However, Vasicek

and Fong [1982] do not propose the specific form of the piecewise polynomial used to model )(~ sZ . The McCulloch and Kochin [2000] method uses a piecewise polynomial to model the logarithmic function of the discount function.

27

3. Estimation Method Selection Criteria and Comparison Results

In this section, we select the estimation method that accurately grasps the characteristics

of the term structure of JGB interest rates from the representative zero curve estimation

methods introduced in Section 2. To those ends, we summarize the JGB interest rate term

structure characteristics in Section 3A. In Section 3B, we set criteria to exclude those zero

curve estimation methods that generate undesirable zero curves as shown in Section 2B and

to select a zero curve estimation method that grasps the JGB interest rate term structure

characteristics. In Section 3C, we select the most appropriate zero curve method from the

representative methods in light of our selection criteria through an analysis based on JGB

market prices.

A. Characteristics of JGB Interest Rate Term Structure

In this subsection, we note two representative characteristics of the term structure of JGB

interest rates from the late 1990s until recently.

The first feature is that the interest rate term structure with remaining maturities of up to

around three years shows a flat curve near zero. Especially during the quantitative easing

policy period from 2001 to 2006 and the global financial crisis since the summer of 2007,

the JGB interest rate term structure generally had that kind of shape. Figure 5 compares the

term structures of U.S. and Japanese government bond interest rates since the financial

crisis. During this period, both countries implemented de facto zero interest rate policies,

but at certain points in time the shapes of the term structures for short-term maturities

differed. In Figure 5(a), the slope of the U.S. government bond yield curve increases from

remaining maturities of two years, while the Japanese curve remains near zero through

remaining maturities of around three years.

28

Figure 5: Government bond interest rate term structures of the U.S. and Japan

(a) June 10, 2010 (b) September 2, 2011

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 4 7 10 13 16 19 22 25 28

Japan U.S.years

%

0

0.5

1

1.5

2

2.5

3

3.5

4

1 4 7 10 13 16 19 22 25 28

Japan U.S.years

%

Note 1: Bloomberg ticker GJGBn Index for Japanese n year interest rate ( 30,20,15,10,,2,1 ��n ). Note 2: Bloomberg ticker USGGn Index for U.S. n year interest rate of two years or more

( 30,10,7,5,3,2,1�n ) and Bloomberg ticker USGG12M Index for one-year interest rate. Source: Bloomberg.

When estimating the zero curve from JGB market prices, an undesirable method may fail

to accurately grasp the curve characteristics introduced above. For example, a portion of the

estimated zero curve for short maturities might fall below zero. As shown in Section 2D,

the polynomial and parsimonious function methods express the entire zero curve as a

polynomial or a specific function. Therefore, it may be difficult to fully capture the

above-mentioned characteristics of the JGB term structure due to the low locality of these

methods.

In recent years, not only the JGB interest rate term structure, but government bond

interest rate term structures in the U.S. and Europe have also been flattening near zero for

short-term maturities. Looking at the U.S. interest rate term structure in Figure 5(b),

compared with Figure 5(a), the term structure is turning flatter near zero through maturities

up to around three years. For this reason, at present, estimations with low locality may not

have sufficient expressiveness to estimate the zero curves in the U.S. and European

countries.14 14 In the U.S., the Federal Reserve Board’s (FRB’s) economists estimate the zero curve based on the method in Svensson [1995] (see Gürkaynak, Sack and Wright [2007] for the details of the estimation method). Our calculations of the zero yield for short-term maturities of one or two months with the zero curve estimation parameters released on the FRB homepage frequently resulted in estimates below zero during periods when the yield curve becomes steep under the low interest rate environment since the middle of 2009.

29

The second feature frequently seen in the JGB interest rate term structure is that it has a

complex shape with multiple inflection points. For example, as in Figure 6, the seven-year

interest rate sometimes becomes relatively low compared with the six-year and eight-year

rates.15

Figure 6: JGB interest rate term structure (February 17, 2009)

0

0.5

1

1.5

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20years

%

Note: n year interest rate – Bloomberg ticker GJGBn Index. Source: Bloomberg

Zero curve estimation methods with a low degree of freedom cannot capture curves with

this kind of complex shape. When the degree of freedom of the estimation method is too

high, however, it may estimate a zero curve with excessive unevenness as seen in Section

2B. Consequently we need an estimation method with a suitable degree of freedom that is

neither too low nor too high, so that it can grasp such characteristics of the JGB interest rate

term structure. In addition, an estimation method that has a large influence on the estimates

for other maturities to capture such uneven shapes is undesirable. Therefore, we believe that

the zero curve estimation method has to have high locality in order to grasp the

characteristics of the JGB interest rate term structure, which has a complex shape with

multiple inflection points.

15 The maturity of the cheapest-to-deliver of the JGB futures is around 7 years. Thus, the prices of JGBs with remaining maturities of around 7 years have often been influenced by the prices of JGB futures. In particular, when the prices of JGB futures rose substantially with the flight to quality since the second half of 2008, JGBs with remaining maturities of around seven years were traded at a higher price than those with remaining maturities of six and eight years. In other words, the 7 year interest rate was lower compared with interest rates of other maturities.

30

B. Estimation Method Selection Criteria

In this subsection, we set criteria to exclude methods which estimate undesirable zero

curves as shown in Section 2B and select methods which capture features of the JGB

interest rate term structure shown in A above.

Criterion (1): Zero curve estimates should not fall below zero

As shown in Section 3A, the JGB interest rate term structure often takes a flat shape near

zero for short-term maturities. Some estimation methods cannot capture this shape, and

may estimate the zero curve below zero for some maturities. As explained in Section 2B,

zero curves that fall below zero for some maturities are undesirable. Accordingly, we

exclude estimation methods that estimate a comparatively large number of zero yield

estimates for remaining maturities of 0.5, 1, 1.5 and 2 years falling below zero during the

estimation period.

Criterion (2): Zero curve estimates should not include abnormal values

As shown in Section 2B, some estimation methods may estimate the zero curve with

abnormal values such as extremely high or extremely low estimates. These methods end up

with under-fitting to JGB market prices due to their low degree of freedom, or get into

over-fitting due to their high degree of freedom. We estimate the zero yield with a specific

maturity using each estimation method on each estimation date; in addition, we calculate

the standard deviation of those zero yield estimates. Based on this, we regard estimates

outside the range of ±2 standard deviations as abnormal values. We then reject estimation

methods with a relatively high frequency of abnormal values. This selection criterion

excludes estimation methods with degrees of freedom that are either too high or too low.

Criterion (3): Theoretical prices should have a good fit with market prices

As noted in Section 3A, the JGB interest rate term structure often has a complex shape

with multiple inflection points. We judge the extent of the fit of the theoretical prices to the

market prices by whether this complex shape is accurately grasped. Specifically, we

31

evaluate it using the residual sum of squares of errors of the market prices and the

theoretical prices on the estimation date. This criterion is used to reject estimation methods

that have poor fits to the market prices. While it does not reject estimation methods that

have over-fits to the market prices, those methods can be rejected by criteria (2) above and

(4) below.

Criterion (4): The shape of curve should not be extremely uneven

As stated in Section 2B, it is undesirable for the zero curve to be extremely uneven. If the

zero yield estimates include abnormal values, there may be large rises and falls in the zero

curve. Those are rejected by criterion (2). Estimation methods that are not rejected by

criterion (2), however, may include methods which estimate zero curves with extreme

unevenness. Thus, in order to exclude this possibility, we check whether or not the

curvature calculated using the zero yield estimates for specific maturities ranging from 0.5

to 20 years in half-year increments from Equation (42) shows excessive values .

.)))1(5.0()5.0(2))1(5.0((39

2

2��

����j

jyjyjy (42)

In this paper, we exclude those estimation methods that frequently generate undesirable

zero curves in light of criteria (1) and (2) from the selection candidates. The selection

results are presented in subsection C(3). Next, the most desirable estimation method is

selected based on the perspectives in (3) and (4) from among those estimation methods that

are not excluded in the above process.

C. Comparative Analysis of Estimation Methods

In this subsection, we select the most suitable estimation method for JGB price data from

the representative zero curve estimation methods in Section 2 in light of the criteria

presented in Section 3B.

(1) Outline of the JGB Price Data Used in the Estimations

32

In our comparative analysis, we use price data on fixed coupon-bearing JGBs (2-year,

5-year, 10-year, 20-year, and 30-year bonds). The price data are Japan Bond Trading Co.,

Ltd. JGB closing prices obtained from the NEEDS provided by Nikkei Digital Media Inc.

The zero curve estimation period covers all business days from January 4, 1999 through

December 30, 2010.

Figure 7: Number of issues

Number of issues of each bond Number of issues of all bonds

0

20

40

60

80

100

120

99/1 00/1 01/1 02/1 03/1 04/1 05/1 06/1 07/1 08/1 09/1 10/1

2-year bond 5-year bond 10-year bond

20-year bond 30-year bond

year/month

0

50

100

150

200

250

300

350

99/1 00/1 01/1 02/1 03/1 04/1 05/1 06/1 07/1 08/1 09/1 10/1year/month

Figure 7 presents the number of fixed coupon-bearing JGB issues over the estimation

period. As shown in the figure, the total number of issues was around 150 in January 1999,

while it has grown to about 300 recently. This is a result of increased issuance of five-year,

20-year and 30-year bonds.

In this way, the JGB issuance conditions have changed since the 2000s. In previous

studies and surveys on the estimation of the JGB zero curve (Komine, Yamagishi et al.

[1989], Oda [1997], Inui and Muromachi [2000], Kawasaki and Ando [2005]), price data

before 2000 are used. In contrast, the estimation results in this paper take into consideration

the change in issuance conditions from the 2000s described above.

(2) Estimation Methods for Our Analysis

The estimation methods for our analysis are the eight methods introduced in Section 2D.

In this subsection, we present supplementary notes on each method.

i. The McCulloch [1975] Method Modeling Discount Rates in Section 2D(1)(i)

The knot points are set in annual increments ranging from year 0 through year 30.

33

ii. The Steeley [1991] Method Modeling Discount Rates in Section2D(1)(ii)

The knot points are set in annual increments ranging from year -3 through year 33.

iii. The McCulloch [1971] Method Modeling Instantaneous Forward Rates in Section

2D(1)(iii)1

The knot points are set in annual increments ranging from year 0 through year 30.

iv. The Steeley [1991] Method Modeling Instantaneous Forward Rates in Section

2D(1)(iii)2

The knot points are set in annual increments from year -2 through year 32.

v. The Tanggaard [1997] Method in Section 2D(2)(i)

The discount rates corresponding to times until when cash flows are not paid cannot be

directly estimated from this model. Hence, this paper estimates those yields by linear

interpolation.

vi. The Schaefer [1981] Method in Section 2D(3)(i)

The Bernstein polynomial is of the fifth degree.

vii. The Nelson and Siegel [1987] Method in Section2D(4)(i)

The Nelder-Mead method is used in the parameter estimation.

viii. The Svensson [1995] Method in Section 2D(4)(ii)

Same as vii.

(3) Comparison Results: Exclusion of Undesirable Estimation Methods

We now summarize the results of comparing the eight methods in light of the criteria (1)

and (2) presented in subsection 3B.

(i) Number of Times That Zero Yield Estimates Fall Below Zero

Table 1 shows the number of times that the zero yield estimates for remaining maturities

of 0.5, 1, 1.5 and 2 years drop below zero during the estimation period under each method.

The table shows that while the piecewise polynomial methods do not give any estimates

below zero, the other methods do give estimates below zero. In particular, the polynomial

method and the parsimonious function methods generate far larger numbers of estimates

34

below zero compared with the other methods.

Table 1: The number of times that interest rate estimates are less than zero Piecewise polynomial methods

Methods modeling discount rates Methods modeling instantaneous forward rates

McCulloch [1975]

Steeley [1991]

McCulloch [1971]

Steeley [1991]

0 0 0 0

Polynomial method Non-parametric method Parsimonious function methods

Schaefer [1981]

Tanggaard [1997]

Nelson and Siegel [1987]

Svensson [1995]

1,084 30 2,162 619 Note: Total of 11,788 samples (4 samples/day × 2,947 days).

Figure 8 presents examples of zero curves on dates when estimates actually fell below

zero for estimation methods with many estimates falling below zero. As shown in Figure 8,

the polynomial method and the parsimonious function methods generate frequent

estimation results with the zero yields below zero for certain maturities during the

quantitative easing policy period. This is considered to result from the low locality of these

methods as explained in Section 3A. Estimates below zero were also observed under the

non-parametric method.

Figure 8: Examples of zero yield estimates below zero

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Nelson and Siegel[1987] (05/5/2) Svensson[1995] (05/8/29)

Tanggaard[1997] (99/3/19) Schaefer[1981](06/2/2)

%

years

Given these results, we conclude that the polynomial method (the Schaefer [1981]

method), the parsimonious function methods (the Nelson and Siegel [1987] method, and the

35

Svensson [1995] method) and the non-parametric method (the Tanggaard [1997] method)

are not appropriate as methods to capture the characteristics of the JGB interest rate term

structure.

(ii) Number of Times That the Zero Yield Estimates Take Abnormal Values

Table 2 summarizes the number of times that the zero yield estimates for each maturity

take abnormal values during the estimation period under each method.

Table 2: The number of abnormalities

Piecewise polynomial method

Polynomial method

Non-parametric method

Parsimonious function

method Method modeling discount rates

Method modeling instantaneous forward rates

Remaining Maturity

McCulloch [1975]

Steeley[1991]

McCulloch [1971]

Steeley [1991]

Schaefer [1981]

Tanggaard [1997]

Nelson & Siegel [1987]

Svensson [1995]

1 year 0 0 2 2 36 11 0 88

2 years 0 0 0 0 0 2 5 0

3 years 0 0 0 0 0 0 101 0

4 years 0 0 0 0 0 0 15 0

5 years 0 0 0 0 0 0 0 0

6 years 0 0 0 0 0 0 0 0

7 years 0 0 0 0 0 0 0 0

8 years 0 0 0 0 0 0 0 0

9 years 0 0 0 0 0 0 0 0

10 years 0 0 0 0 0 0 0 0

11 years 0 0 0 0 0 0 0 0

12 years 0 0 0 0 0 0 0 0

13 years 0 0 0 0 0 0 0 0

14 years 0 0 0 0 0 0 0 0

15 years 0 0 0 0 0 0 0 0

16 years 0 0 0 0 0 0 0 0

17 years 0 0 0 0 0 0 0 0

18 years 0 0 0 0 0 0 0 0

19 years 0 0 0 0 0 0 0 0

20 years 0 0 0 0 0 0 0 0

Notes: 1. There are 2,947 samples for each item (Jan. 4, 1999 – Dec. 30, 2010). 2. The mean and the standard deviation are calculated from the estimates generated by the eight estimation methods at each estimation date, and estimates outside the range of ±2 standard deviations of the mean are defined as abnormal values.

36

The table shows that for short-term maturities, the polynomial method (the Schaefer

[1981] method) and the parsimonious function methods (the Nelson and Siegel [1987]

method, the Svensson [1995] method) give more abnormal values compared with the other

estimation methods. Abnormal values under these methods are caused as a result of

estimates below zero or estimates with far larger values than those of other methods.

Figure 9 is an example of when the estimates for short-term maturities based on the

polynomial method and the parsimonious method are higher than those of other methods.

For a remaining maturity of 0.5 years, the zero yield estimate based on the Schaefer [1981]

method is higher than the estimates based on the other methods.

The reason why the zero yield estimates for short-term maturities under the polynomial

method and the parsimonious function method take abnormal values is that these methods

cannot capture the flat shape of the JGB yield curve near zero for short-term maturities

often seen during the period of quantitative easing policy due to their low locality.

Figure 9: Examples of zero yield estimates with abnormal values

-0.5

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Steeley[1991] modelling discount rate McCulloch[1987] modelling discount rate

Svensson[1994] Shaefer[1981]

%

years

In light of whether the zero yield estimates take abnormal values, we conclude that the

polynomial method and the parsimonious function methods are not appropriate for JGB

zero curve estimates.

Given the above results, we conclude that aside from the McCulloch [1975] Method

Modeling Discount Rates, the Steeley [1991] Method Modeling Discount Rates, the

McCulloch [1971] Method Modeling Instantaneous Forward Rates, and the Steeley [1991]

37

Method Modeling Instantaneous Forward Rates, the other estimation methods do not meet

the selection criteria. Thus, we exclude methods except the above four methods from our

analysis.

(4) Comparison Results: Selection of the Optimal Estimation Method

In this subsection, we select the most appropriate estimation method from the four

remaining methods (the McCulloch [1975] Method Modeling Discount Rates, the Steeley

[1991] Method Modeling Discount Rates, the McCulloch [1971] Method Modeling

Instantaneous Forward Rates, and the Steeley [1991] Method Modeling Instantaneous

Forward Rates) from the perspectives of the fit of theoretical prices to market prices and the

low unevenness of the zero curve.

(i) Fit of Theoretical Prices to Market Prices

Table 3 compares the residual sum of squares under the four estimation methods during

the estimation period. The table indicates that the residual sum of squares of the Steeley

[1991] Method Modeling Discount Rates has the lowest mean value, standard deviation and

maximum value among the four methods during the estimation period.

It appears that this result is because the Steeley [1991] Method Modeling Discount Rates

has the highest locality among these four methods.

Table 3: Comparison Results for Fitness to Market Prices Piecewise polynomial methods

Methods modeling discount rates

Methods modeling instantaneous forward rates

McCulloch [1975]

Steeley [1991]

McCulloch [1971]

Steeley [1991]

Maximum 29.178 27.058 29.126 29.116 Minimum 0.185 0.144 0.198 0.182

Mean 3.414 3.093 3.598 3.807 Standard deviation

4.728 4.441 4.714 5.041

Notes: 1. Mean, standard deviation, maximum and minimum denote the mean value, standard deviation, maximum value and minimum value of the residual sum of squares on each estimation date.

2. The unit is the square of yen.

38

ii Unevenness of Zero Curves

Table 4 summarizes the averaged value of the curvature measuring the extent of the

unevenness of the zero curve during the estimation period. While the table does not show

great differences among the methods, the curve estimated based on the Steeley [1991]

Method Modeling Discount Rates has the lowest curvature.

Table 4: Comparison of Curvatures

Piecewise polynomial functions Methods modeling discount rates Methods modeling instantaneous

forward rates McCulloch

[1975] Steeley [1991]

McCulloch [1971]

Steeley [1991]

1.587×10-2 1.470×10-2 1.930×10-2 1.991×10-2 Notes: 1 The above figures show the mean values of the curvatures (Equation (42)) on each estimation

date during the estimation period. 2 The unit is the square of percent.

Figure 10: The zero curve using the Steeley [1991] Method Modeling Discount Rates

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

%

years Note: Zero curve estimated on February 8, 2000, when the zero curve under the Steeley [1991] Method

Modeling Discount Rates had its highest curvature during the estimation period.

Figure 10 shows the estimation results on the date (February 8, 2000) when the zero

curve under the Steeley [1991] Method Modeling Discount Rates has its highest curvature

during the estimation period. In this figure, the zero yield with a maturity of 9.5 years is

lower than those with maturities of both 9 years and 10 years. This is considered to lead to

39

the increase in the curvature of the zero curve. This reflects the fact that the price of the

JGB with a remaining maturity of about 9.5 years was actually higher than the prices of the

9- and 10-year JGBs; hence, the interpolation is deemed to be appropriate. Considering this,

we conclude that the unevenness of zero curves under the Steeley [1991] Method Modeling

Discount Rates is appropriate.

Summarizing all the above results, among the eight methods considered in our analysis,

the Steeley [1991] Method Modeling Discount Rates is judged the most appropriate for

estimating the JGB market zero curve. In the next section, we clarify the characteristics of

this method through comparisons with the other estimation methods.

4. Characteristics of the Steeley [1991] Method Modeling Discount Rates

In Section 3, the Steeley [1991] Method Modeling Discount Rates (hereafter, “the

Steeley model”) was selected as the optimal estimation method in light of the criteria set in

this paper. In this section, we first clarify the characteristics of the Steeley model in

subsection 4A by comparing the Steeley model with the Nelson and Siegel [1987] Method

(hereafter, “the NS model”). Specifically, we compare the two models’ results in terms of

(1) the shape of curves, (2) the short-term interest rates, and (3) the long-term interest rates.

Next, in subsection 4B, we consider the curve based on the smoothing spline method and

compare it with the curve based on the Steeley model.

A. Comparison of the Steeley Model with the NS Model

(1) Comparison of Zero Curve Shapes

Figure 11 presents the zero curves on December 30, 2010 estimated based on the Steeley

model and the NS model. In Section 3A, we pointed out the flat shape of the curve near

zero for short-term maturities as one of the features of the JGB interest rate term structure.

The curve estimated based on the Steeley model takes that type of shape.

40

Figure 11: Comparison of curve shapes (December 30, 2010)

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Steeley model NS model

%

years

Figure 12 presents the comparison of the two models’ curves on August 20, 2008. Around that time, against the backdrop of a rise in the price of JGB futures, the prices of JGBs with remaining maturities of around seven years rose relative to those of other JGBs. This means that the seven-year interest rate was low compared with the six- and eight-year interest rates. In fact, the zero curve estimated by the Steeley model captures this kind of yield curve shape. In contrast, the zero curve estimated by the NS model does not capture this unevenness from the six-year through the eight-year interest rates. It seems that the Steeley model is able to capture this shape because of its high locality.

Figure 12: Comparison of curve shapes (August 20, 2008)

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Steeley model NS model

%

years

41

year/month

(2) Comparison of Short-Term Interest Rates

Figure 13 presents the short-term interest rate estimates using the Steeley model and the

NS model.

Figure 13: Comparison of short-term interest rates (6-month yields)

-1

-0.5

0

0.5

1

1.5

99/1 00/1 01/1 02/1 03/1 04/1 05/1 06/1 07/1 08/1 09/1 10/1

Steeley model NS model uncollateralized overnight call rate

%

Looking at the figure, the six-month zero yield estimates under the NS model frequently

fall below zero during the quantitative easing policy period. In contrast, the six-month zero

yield estimates during this period based on the Steeley model remain steady near zero,

without falling below zero16.

During periods other than the quantitative easing policy period as well, estimates based

on the Steeley model remain steady without widely diverging from the movements of the

uncollateralized overnight call rate, unlike the estimates using the NS model. In this way,

the Steely model is deemed to steadily estimate short-term interest rates, especially

short-term interest rates during monetary easing periods in Japan.

16 The 6-month zero yields in this period using the Steeley model are a little higher than the zero yields derived from the short-term discount bonds issued by the Japanese government. This seems to reflect that zero curves in this paper are estimated with only JGBs and our estimates of maturities of less than one year include some spreads compared to the zero yields of short-term discount bonds. In this paper, we do not use short-term discount bond prices for the zero curve estimation because the market participants of the short-term discount bond market are different from those of the JGB market.

42

(3) Comparison of Long-term Interest Rates

Figure 14 presents the long-term interest rate estimates (10-year zero yield and 20-year

zero yield) using the Steely model and the NS model.

Figure 14: Comparison of long-term interest rate estimates

10-year zero yield 20-year zero yield

0

0.02

0.04

0.06

0.08

0.1

0.12

0

0.5

1

1.5

2

2.5

3

99/1 00/1 01/1 02/1 03/1 04/1 05/1 06/1 07/1 08/1 09/1 10/1

Deviation (right-hand scale)Steeley modelNS model

year/month

% %

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.5

1

1.5

2

2.5

3

3.5

99/1 00/1 01/1 02/1 03/1 04/1 05/1 06/1 07/1 08/1 09/1 10/1

Deviation (right-hand scale)Steeley modelNS model

year/month

% %

Comparing the estimates using the two models, we find that the largest divergence is

only around 10bp for the 10-year zero yield. For the 20-year zero yield as well, the largest

divergence is also around 10bp. This suggests that there is no substantial difference

between the two models in the estimates of long-term interest rates. However, the selection

of the estimation method is important for analyses using multiple estimates on the curve,

such as analyses using the spread between the short-term and long-term interest rates,

because the short-term interest rate estimates differ depending on the choice of models as

seen in Section 4A(2).

43

B. Comparison of the Steeley Model with Smoothing Spline Methods that Can Estimate

Smooth Instantaneous Forward Rates

An important point to note in using the piecewise polynomial method including the

Steeley model is that the term structure of the instantaneous forward rate may have

excessive unevenness. To overcome this, previous studies focusing on estimations of a

smooth instantaneous forward rates term structure adopt what are known as smoothing

spline methods (Fisher, Nychka and Zervos [1995], Waggoner [1997], Anderson and Sleath

[1999], Jarrow, Ruppert, and Yu [2004], etc.). As seen below, smoothing spline methods

estimate a smooth instantaneous forward rate term structure; nevertheless, the estimates

tend to have lower fits to market prices compared with estimates obtained when smoothing

is not used. As noted above, in this paper, our purpose is to estimate a zero curve that

accurately grasps the features of the JGB yield curve. Accordingly, we do not include

smoothing spline methods in Section 3 because they do not meet the prerequisite of a good

fit to the market prices.

We do compare zero curves and instantaneous forward rate term structures estimated

using the Steeley model and the Jarrow, Ruppert and Yu [2004] model (hereafter, “the JRY

model”), one of the smoothing spline methods.

We now outline the JRY model. In this approach, the instantaneous forward rate is

modeled using a B-spline function based on Equation (21). This method tries to estimate

the zero curve with a good fit to market prices and a smoothed instantaneous forward rate

curve by solving the following optimization problem with a penalty term,

.)')'(())(())((1min0

2T

+,-

���

��� cfT

Idyyf

n)αQPαQP

α (43)

The second term in Equation (43) represents the penalty with respect to the curvature of the

instantaneous forward rate. In other words, the objective function value rises as the

unevenness of the instantaneous forward curve increases. Consequently, the optimal

solution to Equation (43) results in a zero curve estimate with a smooth instantaneous

forward curve. The term ) is called the smoothing parameter. This parameter determines

the unevenness of the instantaneous forward rate term structure. In Jarrow, Ruppert and Yu

44

[2004], this parameter is determined based on the information criteria proposed in Ruppert

[1997].

Figure 15 shows the zero curves and the instantaneous forward rate term structures

estimated with the Steeley model and the JRY model. For the zero curves, the Steeley

model’s curve shows a dip around the maturity of 7 years, while that is not seen in the JRY

model’s curve. This implies that the Steeley model has a good fit to the market prices and

grasps the characteristics of the JGB yield curve. On the other hand, the instantaneous

forward rate curve based on the Steeley model shows great unevenness, while that of the

JRY model is smooth.

Figure 15: Comparison of Steeley model and JRY model estimation results (February 17, 2009)

Zero Curve Instantaneous Forward Rate Curve

0

0.5

1

1.5

2

2.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Jarrow, Ruppert and Yu Steeley

0

1

2

3

4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Jarrow, Ruppert and Yu Steeleyyears years

In the financial business, problems in pricing derivatives underlying the forward interest

rate occur when using an uneven instantaneous forward rate curve. The pricing of the

derivatives is implemented by calculating the future forward curve based on some term

structure model through the initial curve of the instantaneous forward rate. Hence, the price

depends on the initial curve. Therefore, we end up deriving unreasonable prices of those

derivatives if we use an uneven instantaneous forward curve as the initial curve of the term

structure model.

In order to avoid this problem, we can price the derivatives underlying the forward

interest rate with a smooth instantaneous forward curve estimated using a smoothing spline

method. However, as pointed out above, zero curves estimated using smoothing spline

45

methods have a worse fit to market prices. In this way, it is difficult to simultaneously

achieve a good fit to JGB market prices and a smooth instantaneous forward rate term

structure. Thus, we believe that the decision on whether to use the smoothing spline method

or not should depend on how the estimated curve will be used.

5. Conclusion

In this paper, we set criteria to select an appropriate method from among estimation

methods proposed in previous studies in order to grasp the characteristics of the JGB zero

curve, and conducted a comparative analysis of the diverse methods according to the

criteria. Specifically, we excluded estimation methods which did not meet the criteria that

estimates should not fall below zero and take abnormal values. We then selected the most

appropriate estimation method based on the criteria of a good fit to market prices and low

unevenness in the zero curve. As a result, we found that the Steeley [1991] Method

Modeling Discount Rates was the most appropriate in light of the criteria.

The zero yields estimated with the Steeley model did not drop below zero during the

estimation period, and captured the characteristics frequently seen in the JGB yield curves

including the flat shape near zero for short-term maturities and the inflection points in the

curve for mid-term maturities. Zero curves estimated appropriately in this manner are

considered to be important as a starting point for numerous analyses on JGB interest rates.

Since 2010, the U.S. and European government bond yield curves have also been showing

a flat shape near zero for short-term interest rates, coming to resemble the characteristics of

the JGB yield curve during quantitative easing policy periods and in recent years. Thus, the

Steeley model is likely to prove effective for zero curve estimations of the U.S. and

European countries.

In conclusion, we would like to note two outstanding challenges for future research. First,

the analysis in this paper is based on JGB data from the late 1990s through 2010. Hence,

conducting similar analysis using data from a different time or market might lead to

different conclusions. Consequently, one challenge is to conduct a comparative analysis on

zero curve estimation methods using more wide-ranging data. Second, the analysis in this

46

paper uses the simple residual sum of squares of the market prices and theoretical prices as

the objective function in estimating the parameters defining the zero curve. Depending on

the market environment and the purpose of utilization of the zero curve, however, it may be

more appropriate to conduct the estimation based on a different objective function.

Therefore, another challenge for future research is to conduct the analysis using different

objective functions.

47

Appendix 1. Market Conventions for Calculation of JGB Theoretical Prices

To precisely calculate the theoretical prices, it is necessary to understand JGB market

conventions including the definition of the timing of when cash flow is paid, the method of

calculating the number of days until cash flow is paid, and the method of calculating

accrued interest. This appendix explains the items required for a precise calculation of

theoretical prices.17 A. Definition of Terms

Here we present the terms required to explain JGB market conventions.

Interest Payment Dates

The dates on which coupons are paid on a bond are referred to as the interest payment

dates. For fixed-coupon bearing bonds, these are normally set once every half year, such as

“June 20 and December 20 each year.” As explained in Appendix 1-B below, it is necessary

to note that the interest payment dates sometimes differ from the dates on which coupons

are actually paid. The interest payment dates are announced by the Ministry of Finance

when JGB bond auctions are held. The interest payment dates on the five-year, 10-year,

20-year and 30-year bonds are all either March 20 and September 20 or June 20 and

December 20 each year18. The interest payment dates on two-year bonds are the 20th of

each month for bonds issued through September 2007, and the 15th of each month for

bonds issued since October 2007.

Initial Interest Payment Date

The first interest payment date after a given bond is issued, for the initial interest

payment period, is referred to as the initial interest payment date. The initial interest

payment date and the actual interest payment date are sometimes different.

Initial Interest Payment

The initial interest payment is the amount of cash flow obtained when the first interest 17 We referred to Ohta [2003] for an explanation of this Appendix. 18 We here note that some of 30-year bond issues have either February 20 and August 20 or May 20 and November 20 as the interest payment dates.

48

payment occurs. Here we note that this is not the interest payment date.

Prior Term Interest Payment Date

The prior term interest payment date is defined as the date half a year prior to the initial

interest payment date. For example, if the initial interest payment date is June 20, 2008 then

the prior term interest payment date is December 20, 2007.

Trade Date

The trade date is the date on which the buyer and seller agree to the transaction.

Settlement Date

The settlement date is the date three business days after the trade is executed, counting

from the day after the trade date. The settlement date, not the trade date, is used as the

standard for the calculation of occurrence of cash flow and accrued interest.

B. Market Conventions for Calculating the Number of Days until Cash Flow is Paid

Here we explain the JGB market conventions for counting the number of days from the

settlement date until the time when cash flow is paid. This is one of the factors used to

calculate JGB theoretical prices.

First we explain the case when the interest payment date is a business day. In this case

the interest payment date is the date when cash flow is actually paid. With 365 days in a

year, the number of the days from the settlement date to the interest payment date is

counted as the number of days until each interest payment date including leap days for

issues with less than a year remaining until the maturity date. For issues with a year or

longer remaining until the maturity date, the count is the number of days from the

settlement date to each interest payment date deducting any leap days during that interval.

The one-end method is applied to the calculation of the number of days between the

settlement date and interest payment dates. In this method, the settlement date is not

counted in calculation of the number of days, but each of the interest payment dates is

counted. On the JGB market, the one-end method is applied to the day counting for all

49

fixed-coupon bearing bonds.19

In cases when the interest payment date is not a business day, the modified following

business day convention is applied on the JGB market. Under this convention, when the

interest payment date is not a business day and the subsequent business day is in the same

month as the coupon date, the cash flow is paid on that date, and when the subsequent

business day is in the month after the interest payment date, the cash flow is paid on the

business day immediately preceding the interest payment date. As a specific example, many

JGBs had interest payment dates on June 20, 2009. Because that date was a Saturday, the

modified following business day convention was applied and the cash flow was paid on

June 22, 2009.20 In cases when the interest payment date is not a business day, the one-end

method is also applied to the day counts between the settlement date and interest payment

dates.

C. Response to Changes in Legal and Market Systems

Looking over the history of the JGB market, the times when cash flows were paid and

the amount of cash flow had shifted along with changes in market and legal systems. As

examples, we examine (1) the introduction of the immediate reopening rule, (2) the Happy

Monday system, and (3) the settlement system.

(1) Introduction of the Immediate Reopening Rule (Reopening System)

The immediate reopening rule (hereafter, “reopening system”) introduced in March 2001

is a system whereby upon issuance of a new bond with the same face value, coupon rate

and maturity date of an already issued bond, the two bonds are treated as the same issue.

Prior to the introduction of this system, the old and new bonds had to be treated as separate

issues until the initial interest payment date even if the issuance conditions aside from the

19 Prior to the introduction of the reopening system in March 2001 (see C (1) in Appendix 1 below), the both-ends method was used for the calculation of accrued interest. The both-ends method is the method that counts both the settlement date and the coupon date in the day counting. 20 The auction results for past interest-bearing JGBs are presented on the Ministry of Finance web page http://www.mof.go.jp/english/jgbs/auction/past_auction_schedule/auct_resul/index.htm.

50

issuance date were identical, because the initial interest payments were different. The

reopening system was introduced to reduce such complexities of managing issues.

The JGB initial interest payment calculation methods differ before and after the

introduction of the reopening system. The accrued interest calculation methods also differ.

The specific calculation methods are explained below.

(2) The Happy Monday System

The Happy Monday System moves a number of public holidays to Mondays, to create

three-day weekends. This system began under the Act on Partial Revision of the Act on

National Holidays, which came into effect from January 2001. This moved the Coming of

Age Day (January 15) and Health and Sports Day (October 10) holidays to the second

Mondays of their respective months. Then under the Act on Partial Revision of the Act on

National Holidays and the Act on Social Welfare Service for Elderly, which came into

effect from January 2003, the Marine Day (July 20) and Respect for the Aged Day

(September 15) holidays were moved to the third Mondays of their respective months.

The second revision had the greater effect from the perspective of calculating JGB

theoretical prices. For example, July 20, which was a holiday prior to the revision, could

now be a business day and the actual date when two-year JGB cash flow was paid changed

by the revision of the act. In addition, with the revision, September 20 may now be

Autumnal Equinox Day or Respect for the Aged Day. This could result in substitute

holidays when Sundays and holidays overlap, potentially pushing the interest payment

dates and the actual dates when cash flows are paid ahead or behind by even a few days.21

For estimating the zero curve, it is necessary to specify exactly when such system

changes were incorporated into market prices. Yet it is very difficult to accurately specify

such information from the market prices.

For this reason, in our analysis in this paper, when calculating theoretical prices on trade

dates from June 22, 2001 (when the Act on Partial Revision of the Act on National 21 For example, in September 2009, interest payment date was changed from the 21st to the 24th of the month as a result of that Respect for the Aged Day was moved to the third Monday. This is because the weekday between Respect for the Aged Day (September 21) and Autumnal Equinox Day (September 23) became a national holiday under the provisions of the Act on National Holidays.

51

Holidays and the Act on Social Welfare Service for Elderly was published in the

government gazette), the dates when cash flows are paid after January 2003 (when that Act

came into effect) are assumed to be determined based on the revised holidays system.

(3) Settlement System

On the JGB market, settlements among large brokerages switched to T+7 (delivery seven

business days later counting from the day after the trade) from September 19, 1996 and

then to T+3 (delivery three business days later counting from the day after the trade) from

April 1, 1997. Because the calculation of theoretical JGB prices is based not on the trade

date but on the settlement date, changes in the number of days between the trade date and

the settlement date influence the theoretical price calculations. Because the market price

data for our analysis in this paper run from January 1999 through December 2010; however,

the above changes in the settlement system do not affect the JGB theoretical price

calculations herein.

D. Initial Interest Payment and Accrued Interest Calculation Methods

As noted above, the initial interest payment and accrued interest calculation methods

vary before and after the introduction of the reopening system in March 2001. Thus, we

explain the calculation methods before and after the introduction of the reopening system.

(1) Initial Interest Payment Calculation Method

Prior to the introduction of the reopening system, the initial interest payment for each

issue was determined along with the issuance date as described below. As a result, issues

with the same face value, coupon rate and maturity but with different issuance dates had

different initial interest payments.

The initial interest payment ),,( 1iii vvv TNcg before the introduction of the reopening

system is defined below22. When the issuance date of the bond was before the prior term 22 In D (1) and D (2) of Appendix 1, we assume that the prior term interest payment date means the date after applying the modified following business day convention if it is not a business day.

52

interest payment date, the definition was as shown by the following equation with d as

the number of days from the issuance date to the prior term interest payment date using the

both-ends method.

.3652

1),,( 1 ���

� ��

dNcTNcg iiiii vvvvv

When the issuance date of the bond was after the prior term interest payment date, the

definition was as shown by the following equation with d as the number of days from the

prior term interest payment date to the date when the first interest payment occurs using the

both-ends method.

.365

),,( 1dNcTNcg iiiii vvvvv �

For bonds issued since the introduction of the reopening system, the initial interest

payment is defined as one half the product of the coupon rate multiplied by the face value

as shown by the following equation, regardless of the issuance date.

.2

),,( 1

iiiii

vvvvv NcTNcg �

(2) Accrued Interest Calculation Method

When a bond is traded between two interest payment dates, the seller has the right to

receive cash flow corresponding to the number of days from the last day that cash flow was

paid to the settlement date. However, the cash flow for this period is all paid to the bond

holder when the subsequent cash flow is paid. Hence, the buyer pays this cash flow which

the seller should receive by adding it to the market price when the bond is traded. This

amount added is referred to as accrued interest. As mentioned above, on the JGB market the

method of calculating accrued interest changed with the introduction of the reopening

system. Accordingly, the following discussion is divided into the cases before and after the

introduction of the reopening system in March 2001.

First, we present the accrued interest calculation method before the introduction of the

reopening system. When the settlement date was prior to the date when the first interest

payment occurs, the accrued interest ivA was calculated as follows with 'd as the

53

number of days from the issuance date to the settlement date according to the both-ends

method, which includes leap days.

.365

'dNcA iii vvv ��

Before the introduction of the reopening system, when the settlement date was after the

date when the first interest payment occurs, the accrued interest ivA was calculated as

follows with 'd as the number of days from the last date that cash flow is paid to the

settlement date.

.183'

2

183'365

'

���

���

�

.

���

dNc

ddNcA

ii

ii

i

vv

vv

v

The method of calculating accrued interest after the introduction of the reopening system

is divided into the two cases (1) when the settlement date is before the initial interest

payment date and (2) when the settlement date is after the initial interest payment date, as

described below. In case (1), 'd is the number of days from the settlement date to the prior

interest payment date using the one-end method including leap days within the interval, and

in case (2), 'd is the number of days from the last day of occurrence of the cash flow to

the settlement date using the one-end method including leap days within the interval. Then,

the accrued interest is calculated using the following equation for both case (1) and case

(2).

.183"

2

183"365

"

���

���

�

.

���

dNc

ddNcA

ii

ii

i

vv

vv

v

54

Appendix 2. Estimation Algorithms for the Steeley [1991] Method

In Appendix 1, we summarized the market conventions required to calculate JGB

theoretical prices regarding the definition of when cash flow is paid, the method of

calculating the number of days until when cash flow is paid, and the determination of cash

flow amounts (initial interest payments and accrued interest). The theoretical aspects of the

Steeley [1991] model selected in Section 3 are presented in Section 2, but when actually

making an estimate from JGB price data with this method, the calculations must be

consistent with the market conventions explained in Appendix 1. In this Appendix 2 we

now explain the algorithms to efficiently conduct estimations using the Steeley [1991]

model while having the contents explained in Appendix 1 accurately reflected in the JGB

theoretical price calculations.

First we classify all JGBs traded at 0�t into the following four groups.

i. Issues with less than a year until maturity with interest payment dates on the 20th.

ii. Issues with less than a year until maturity with interest payment dates on the 15th.

iii. Issues with a year or more until maturity with interest payment dates on the 20th.

iv. Issues with a year or more until maturity with interest payment dates on the 15th.

Groups ii and iv cover all the two-year bonds issued after October 2007, as shown in

Appendix 1-A. Groups i and iii cover all other bond issues. The reason why we separate

bonds that will mature within one year from all other bonds is that the adjustment of the

leap year to calculate the number of days until the payment of cash flow differs according

to this difference in the remaining maturities (see Appendix 1-B).

For simplicity, we show how to calculate the theoretical bond prices only for group iii

below. First, we set 0�t as a certain day of month W in year V. The maturity date of the

issue with the longest time to maturity among all issues in group iii is assumed to be a

certain day of month Y in year X. Then the number of months between these two dates,

including both months to which both dates belong, is equal to 1)(12 ���� YWVX .

Hereafter this value is referred to as “ monthn .”

Next, we set the date on which issues in group iii can pay cash flow as the 20th of each

month from month W year V to month Y year X in cases when the 20th is a business day,

55

and we set the date as the business day according to the modified following business day

convention in cases when the 20th is not a business day. Then we calculate the number of

years from 0�t to each of those dates based on the number of days calculation method in

Appendix 1-B. We set those as monthnTTT ��� �21 .

Additionally, if issues in group iii at 0�t are expressed as ),1( nivi �� , then the

amount of cash flow for issue iv on the 20th of each month (or on the date determined by

the modified following business day convention in cases when the 20th is not a business

day) from month W year V to month Y year X is determined. We denote these cash flows

for issue iv as T1 )~,,~(:~ i

month

ii vn

vv cc ��c . All JGB issues generate interest payments every

half year; hence, it is important to note that some of elements of ivc~ take zero.

In order to calculate all discount functions until dates when cash flows are paid, we

define matrix B with an element taking the value of a B-spline function for all points of

time below. We define the ),( jk th element of matrix B as ),4( kTjB � where ),( xkB

is calculated from Equation (16) introduced in subsection 2D(1)(ii). In other words, we

define matrix B as shown in Equation (A-1).

.

),1(),3(

),1(),3( 11

���

�

�

�

��

���

monthmonth nknotn

knot

TnBTB

TnBTB

����

�B (A-1)

Here, we find that the discount rate )( jTZ for jT becomes the jth element of Bα from

Equation (15) if we set the parameter of the Steeley [1991] Method as T

13 ),,( ���knotn�� �α .

Thus, the theoretical price of issue iv , )(αivQ is calculated as shown in Equation

(A-2).

.)~()( T Bαcα ii vvQ � (A-2)

The efficiency of this algorithm is that when we calculate the discount function, it is not

necessary to calculate the number of days until each cash flow is paid for each issue and a

B-spline function for each day. In other words, within the same group of issues (the group

iii issues), we have only to calculate the B matrix just once following Equation (A-1).

In the above, we present the calculation algorithm for the theoretical prices of group iii

56

issues. Like the above case, applying the same algorithm for the issues in the other groups

(groups i, ii and iv), we can calculate the theoretical prices following Equation (A-2) by

calculating the B matrix just once for each group from Equation (A-1). Thus, this

algorithm enables us to estimate the zero curve efficiently through calculating the B

matrix for each of the four groups rather than calculating the spline functions for the cash

flow paid at each point in time for each issue.

57

References

Bank for International Settlements, “Zero-coupon Yield Curves: Technical

Documentation,” BIS paper No.25, 2005.

Carleton, Willard T., and Ian A. Cooper, “Estimation and Uses of the Term Structure of

Interest Rates,” Journal of Finance, 31(4), 1976, pp.1067�1083.

Fisher, Mark, Douglas Nychka, and David Zervos, “Fitting the Term Structure of Interest

Rates with Smoothing Splines,” Working Paper, Board of Governors of the Federal

Reserve System, 1995.

Gürkaynak, Refet S., Brian Sack, and Jonathan H. Wright, “The U.S. Treasury Yield Curve:

1961 to the Present,” Journal of Monetary Economics, 54(8), 2007, pp.2291�2304.

Houglet, Michel X., “Estimating the Term Structure of Interest Rates for Non-homogeneous

Bonds,” Ph.D Dissertation, University of California Berkeley, 1980.

Inui, Koji, and Yukio, Muromachi, Kinyu Model ni okeru Suitei to Saitekika (Estimation

and Optimization in Financial Modeling), Asakura Shoten, 2000 (in Japanese).

Ioannides, Michalis, “A Comparison of Yield Curve Estimation Techniques using UK

Data,” Journal of Banking and Finance, 27(1), 2003, pp.1�26.

Jarrow, Robert, David Ruppert, and Yan Yu, “Estimating the Interest Rate Term Strucuture

of Corporate Debt with a Semiparametric Penalized Spline Model,” Journal of the

American Statistical Association, 99(465), 2004, pp.57�//0

Kalev, Petko S., “Estimating and Interpreting Zero Coupon and Forward Rates: Australia,

1992-2001,” Working Paper, Monach University, 2004.

Kawasaki, Yoshinori, and Tomohiro, Ando, “Estimating Term Structure Using Non-linear

Splines: A Penalized Likelihood Approach,” Zerger A. and Argent R. M. eds., Advances

in Applications for Management and Decision Making, 2005, pp.864�870.

Komine, Midori, Masaaki, Yamagishi, Kazuyuki, Matsumoto, Takashi, Futatsugi, Jun,

Tsukasa, Tomoyuki, Nagao, Kazuhiko, Sunagawa, and Naoshi, Sano, “Wagakuni Saiken

58

Shijou no Koyuu no Gensho to Kikan Kouzou Bunseki (Analyses of Japan’s Bond

Market and it’s Term Structure),” Financial Review, 14, Ministry of Finance, 1989 (in

Japanese) .

McCulloch, J. Huston, “Measuring the Term Structure of Interest Rates,” Journal of

Business, 44(1), 1971, pp.19�31.

────, “The Tax-adjusted Yield Curve,” Journal of Finance, 30(3),1975, pp.811�830.

────, and Levis A. Kochin, “The Inflation Premium Implicit in the US Real and Nominal

Term Structures of Interest Rates,” Working Paper, Ohio State University, 2000.

Nelson, Charles R., and Andrew F. Siegel, “Parsimonious Modeling of Yield Curves,”

Journal of Business, 60(4), 1987, pp.473�489.

Oda, Nobuyuki, “A Note on the Estimation of Japanese Government Bond Yield Curves,”

IMES Discussion Paper No.96-E-27, Bank of Japan, 1996.

Ohta, Tomoyuki, Shin Saiken Unyou to Toushi Senryaku (Bond Investment and Investment

Strategy), Kinyu Zaisei Jijo Kenkyukai, 2003 (in Japanese).

Ruppert, David, “Empirical-bias Bandwidths for Local Polynomial Nonparametric

Regression and Density Estimation,” Journal of the American Statistical Association,

92(439), 1997, pp.1049�1062.

Schaefer, Stephen M., “Measuring a Tax-specific Term Structure of Interest Rates in the

Market for British Government Securities,” Economic Journal, 91(362), 1981,

pp.415�438.

Steeley, James M., “Estimating the Gilt-edged Term Structure: Basis Splines and

Confidence Intervals,” Journal of Business Finance and Accounting, 18(4), 1991,

pp.513�529.

Svensson, Lars E. O., “Estimating Forward Interest Rates with the Extended Nelson and

Siegel Method,” Sveriges Riksbank Quarterly Review, 3(1), 1995, pp.13-26.

Tanggaard, Carsten, “Nonparametric Smoothing of Yield Curves,” Review of Quantitative

Finance and Accounting, 9(3), 1997, pp.251�267.

59

Vasicek, Oldrich A., and H. Gifford Fong, “Term Structure Modeling Using Exponential

Splines,” Journal of Finance, 37(2), 1982, pp.339�348.

Waggoner, Daniel F., “Spline Methods for Extracting Interest Rate Curves from Coupon

Bond Prices,” Federal Reserve Bank of Atlanta Working Paper 97-10, 1997.